# If $\det(A+B)$ and $\det(A)$ are real numbers, must $\det(B)$ be real?

Is this true for any two matrices of complex numbers, and if not, what counterexamples are there? If one of these was a matrix of real numbers, would this property hold?

I've been playing around with this, but since I can't find a way to write $$\det(A+B)$$ in terms of $$\det(A)$$ and $$\det(B)$$. I haven't made much progress.

• Regarding writing $\det(A+B)$ in terms of $\det(A),\det(B)$: linear algebra is not my strongest suit, but I believe that this is in general impossible. Certain approximations can be very useful, however, if the appropriate norm of one matrix is small with respect to the other. – FearfulSymmetry Sep 4 '20 at 18:35
• @Kriesler, it is considered poor form on this website to delete or clear your question after you receive an answer. If you're satisfied with the answer provided, you can accept it. – FearfulSymmetry Sep 4 '20 at 19:00

No. Consider $$A=\pmatrix{1&1\\ 0&1}$$ and $$B=\pmatrix{0&-1\\ i&0}$$ for instance.